I want to solve a optimal control problem with the Pontryagin-Maximum-Principle, given following differential equation:
$\dot x(t)=(a+b-c(t))x-c(t)\\ x(0)=x_0$
Regarding to the conditions: $x\geq0$ maximizing $U_{Ges}=\int_{0}^{\infty} e^{-rt} U(c(t))dt$
How can i derive the following transversality conditions? $\lim\limits_{t \rightarrow \infty}{e^{-rt}p(t)}\geq 0 \\ \lim\limits_{t \rightarrow \infty}{e^{-rt}p(t)x(t)}=0$
I set up all necessary condition for finding an optimal control c(t) but cannot find any derivation of the transversality conditions for infinite time horizons like that.